Page 21 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

Chapter 2
A Progressive Asymptotic Approach Procedure

for Simulating Steady-State Natural Convective
Problems in Fluid-Saturated Porous Media

In a fluid-saturated porous medium, a change in medium temperature may lead to a
change in the density of pore-fluid within the medium. This change can be consid-
ered as a buoyancy force term in the momentum equation to determine pore-fluid
flow in the porous medium using the Oberbeck-Boussinesq approximation model.
The momentum equation used to describe pore-fluid flow in a porous medium is
usually established using Darcy’s law or its extensions. If a fluid-saturated porous
medium has the geometry of a horizontal layer, and is heated uniformly from the
bottom of the layer, then there exists a temperature difference between the top
and bottom boundaries of the layer. Since the positive direction of the tempera-
ture gradient due to this temperature difference is opposite to that of the gravity
acceleration, there is no natural convection for a small temperature gradient in the
porous medium. In this case, heat energy is solely transferred from the high tem-
perature region (the bottom of the horizontal layer) to the low temperature region
(the top of the horizontal layer) by thermal conduction. However, if the temperature
difference is large enough, it may trigger natural convection in the fluid-saturated
porous medium. This problem was first treated analytically by Horton and Rogers
(1945) as well as Lapwood (1948), and is often called the Horton-Rogers-Lapwood
problem.
This kind of natural convection problem has been found in many geoscience
fields. For example, in geoenvironmental engineering, buried nuclear waste and
industrial waste in a fluid-saturated porous medium may generate heat and result
in a temperature gradient in the vertical direction. If the Rayleigh number, which
is directly proportional to the temperature gradient, is equal to or greater than the
critical Rayleigh number, natural convection will take place in the porous medium,
so that the groundwater may be severely contaminated due to the pore-fluid flow cir-
culation caused by the natural convection. In geophysics, there exists a vertical tem-
perature gradient in the Earth’s crust. If this temperature gradient is large enough,
it will cause regional natural convection in the Earth’s crust. In this situation, the
pore-fluid flow circulation due to the natural convection can dissolve soluble min-
erals in some part of a region and carry them to another part of the region. This is
the mineralization problem closely associated with geophysics and geology. Since a
natural porous medium is often of a complicated geometry and composed of many

C. Zhao et al., Fundamentals of Computational Geoscience, 7
Lecture Notes in Earth Sciences 122, DOI 10.1007/978-3-540-89743-9 2,
C Springer-Verlag Berlin Heidelberg 2009